{
  "annotation_id": "3e4ff1d4-f560-4fd8-ac00-a6e9039b5a20",
  "date_created": "2026-02-17T05:53:12.484000Z",
  "date_modified": "2026-02-17T05:53:12.484000Z",
  "file_hash": "8bfeecb2fef6a0c0bb2d5102ff041aba0d1a89f64e785ed43aaf664f6162f931",
  "private": false,
  "record": {
    "abstract": "We study completion of partial matrices with nonnegative entries to matrices of nonnegative rank at most $r$ for some $r \\in \\mathbb{N}$. Most of our results are for $r \\leq 3$. We show that a partial matrix with nonnegative entries has a nonnegative rank-1 completion if and only if it has a rank-1 completion. This is not true in general when $r \\geq 2$. For $3 \\times 3$ matrices, we characterize all the patterns of observed entries when having a rank-2 completion is equivalent to having a nonnegative rank-2 completion. If a partial matrix with nonnegative entries has a rank-$r$ completion that is nonnegative, where $r \\in \\{1,2\\}$, then it has a nonnegative rank-$r$ completion. We will demonstrate examples for $r=3$ where this is not true. We do this by introducing a geometric characterization for nonnegative rank-$r$ completion employing families of nested polytopes which generalizes the geometric characterization for nonnegative rank introduced by Cohen and Rothblum (1993).",
    "arxiv_id": "2601.07658",
    "authors": [
      "Kaie Kubjas",
      "Lilja Mets\u00e4lampi"
    ],
    "categories": [
      "math.MG",
      "math.AG",
      "math.CO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "Geometry of low nonnegative rank matrix completion",
    "url": "https://arxiv.org/abs/2601.07658",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "ffb68794-ec57-403f-be4c-8eb63547626a",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}